AUTOMATION IN HOWITZER

A MAJOR PROJECT REPORT

SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE

AWARD OF

THE DEGREE OF

BACHELOR OF ENGINEERING (Instrumentation & Control Engineering)

SUBMITTED TO

PUNE UNIVERSITY

SUBMITTED BY Name of Student University Seat No.

Gunjal Gajanan B80784628

Bobhate Rohan B80784649

Kumbhare Prathmesh B80784632

GUIDED BY

Professor N.M. Karajanagi HOD – Instrumentation & Control

May 2014

(Month & Year of Submission)

GOVERNMENT COLLEGE OF ENGINEERING AND RESEARCH

AWASARI (KHURD)

TABLE OF CONTENTS

Page No.

Acknowledgement i

Abstract ii

List of Tables iii

List of Figures iv

List of Abbreviations v

Page No.

Chapter 1: INTRODUCTION 1

Chapter 2: LITERATURE REVIEW 3

2.1 Howitzer introduction 3

2.2 Earlier firing methods 6

2.3 Projectile and shell types 12

2.4 Point mass model 18

2.5 Ballistics coefficient, Drag coefficient and Mach number 21

2.6 Fuzzy logic basics and fuzzy systems 26

Chapter 3: PRESENT WORK 4

3.1 Block diagram 31

3.2 MATLAB algorithm 32

3.3 Hardware implementation for the project 40

Chapter 4: Results And Discussion 43

Chapter 5: Conclusion And Future Scope 46

References 47

iii

List of Tables

Table Title Page

1 Various types of projectile 13

2 Types of shells & its uses 17

iv

LIST OF FIGURES

Figure Title Page

1 HOWITZER image 5

2 Drag coefficient 22

3 Mach number 25

4 Steps in fuzzy system 30

5 Block diagram of model 31

6 Fuzzy Toolbox 36

7 Actual signal conditioning 39

8 Model of HOWITZER 41

9 Firing table of 155mm projectile 44

v

ABBREVIATION

FO Forward observer

FDC Fire direction center

BRL Ballistics research laboratory

ii

Abstract

Howitzer cannons form a vital part in the war artillery in modern defense of our country. The

firing techniques employed in the past were time consuming and a bit tedious. With the

introduction of modern technologies this process has been considerably improved with

increased accuracy. This project report introduces another novel approach towards the firing

technique of Howitzer cannons. This approach demonstrates increased simplicity in designing

of firing algorithm and also improvement in the time required for firing. Another important

point worth mentioning is this report mainly dwells on the software algorithm developed for the

firing process and gives the mechanical part (hardware) secondary importance. The algorithm

which has been constructed finds its roots in the fuzzy logic. The firing process becomes

relatively easier as the fuzzy logic dictates this algorithm. Computation speed increases and so

ultimately the rate of firing improve. Incorporation of software was necessary; the algorithm

takes help of MATLAB for fuzzy logic implementation and MULTISIM for electrical circuits.

The hardware part (model) is a miniature of Howitzer cannon which is capable of both the

movements i.e. the vertical and horizontal rotation. Our model utilizes the pneumatic energy to

power the firing projectile. Projectile motion is an active research area and its accuracy is

constantly improving but for the sake of simplicity point mass model differential equations and

its assumptions were considered. In the latter stages of the report the readings and calculations

of the trajectories are included and finally possible future additions are mentioned.

ACKNOWLEDGEMENT

I would like to place on record my deep sense of gratitude to Prof. N.P. Futane HOD-Dept. of

Electronics & Telecommunications Engineering,. Government College Of Engineering And

Research, Awasari(Khurd), for his generous guidance, help and useful suggestions.

I express my sincere gratitude to Prof N.M. Karajanagi HOD-Dept. ofInstrumentation & Control

Engineering, Government College Of Engineering And Research, Awasari(Khurd), for his

stimulating guidance, continuous encouragement and supervision throughout the course of present

work.

I also wish to extend my thanks to Prof. N.P. Wagh and other colleagues for attending my seminars

and for their insightful comments and constructive suggestions to improve the quality of this

project work.

Signature(s) of Students

~n Kumbhare PrathrneshGunjal Gajanan

(B80784628) (B80784649) (B80784632)

CERTIFICATE

I hereby certify that the work which is being presented in the B.E. Major Project Report entitled

"AUTOMATION IN HOWITZER", in partial fulfillment of the requirements for the award of

the Bachelor of Engineering in Instrumentation & Control Engineering and submitted to the

Department of Instrumentation & Control Engineering of Government College of Engineering

& Research,Awasari (Khurd) is an authentic record of my own work carried out during a period

from July 2013 to June 2014 under the supervision of Prof. N.M. Karajanagi, HOD-

Instrumentation & Control Department.

The matter presented in this Project Report has not been submitted by me for the award of

any other degree elsewhere.

~.Gunjal Gajanan

(880784628)

Signature of Student

B£:(880784649)

Kumbhare Prathmesh

(880784632)

This is to certify that the above statement made by the student(s) is correct to the best of my

knowledge.

Signature of

External Examiner

Signature of

Principal

Prof. .

Project Guide

jana~/

Head of Department

1

Chapter 1

Introduction

This section introduces to both the methods of firing the projectile. The first discussed method

is the traditional one which was employed earlier in the warfare while the other method is

modern one where fuzzy logic is utilized.

In the early stages the distance at which the target is located was given by one person

called as forward observer. This person who was situated in the nearby area would compute the

distance at which the target is located with the help of binoculars. Once the distance was

determined, he would give this value to the operating staff of the cannon. Depending upon the

distance the cannon would be adjusted and then fired. The movement of the cannon by done

manually by the soldiers in the beginning but today the motors and gears drives the cannon.

This method makes use of something called as firing table during firing. The firing table is

nothing but a database created by taking tests of the cannon. This table accounts for wind

velocity, air temperature and atmospheric pressure correction factors in the trajectory

determination. So each time the forward observer gives distance of the target, using firing table

data the soldiers set the cannon accordingly. This procedure is described with the following

image.

2

Figure 1 Earlier Firing Procedure

The second method which is introduced is the modern firing approach which uses fuzzy logic

knowledge. The need for this method is clear as the earlier method states that the forward

observer gives information to the control center until the projectile hits the exact desired

location. This shows that the method is time consuming and of lesser accuracy.

Modern approach to this process eliminates all the complexity involved. Once the distance

of the target is acquired, the algorithm computes the angle required for corresponding distance

taking into account the factors mentioned above. The software used for angle determination is

based on the fuzzy logic rules. These rules are simple IF-THEN rules which create the

membership functions forming the fuzzy system. Once the system gives computed angle of

firing then electronic circuitry is employed to drive the mechanical assembly which adjusts the

barrel of the cannon. The electronic circuitry is basic signal conditioning circuit which converts

the angle value into suitable electronic signal to drive gear and motor assembly. Results taken

using this method shows that it is easier for computation and takes less time for firing.

3

Chapter 2

Literature Review

Information given in this section is a compilation of all the data that forms the core for this

project. Methodology of the earlier firing method is given in brief. Projectile shell

specifications and its various types depending upon many factor is presented next. As the

projectile interacts in the non-vacuum model, environmental factors must be addressed and so a

realistic model must be analyzed. Various forces that act on the projectile during motion are

explained and suitable assumptions are made. Differential equations are required to compute

and explain the projectile motion in non-vacuum model. Depending upon the accuracy required

and factors to be considered, four major different sets of differential equations are used for

computation. These equations are presented and model which is used for this project i.e. point

mass model in given in detail. In the last part the basic rules and theory of fuzzy logic is

presented with the steps required for constructing fuzzy systems.

2.1 Howitzer introduction:

Information about Howitzer from Wikipedia pages is collected and presented below:

A howitzer is a type of artillery piece characterized by a relatively short barrel and the use of

comparatively small propellant charges to propel projectiles at relatively high trajectories, with

a steep angle of descent. Until fairly recently, about the end of the Second World War, such

weapons were characterized by a barrel length 15 to 25 times the caliber of the gun.

In the taxonomies of artillery pieces used by European (and European-style) armies in the 17th,

18th, 19th, and 20th centuries, the howitzer stood between the "gun" (characterized by a longer

barrel, larger propelling charges, smaller shells, higher velocities, and flatter trajectories) and

the "mortar" (which was meant to fire at even higher angles of ascent and descent). Howitzers,

like other artillery pieces, are usually organized in groups called batteries.

The English word howitzer comes from the Czech word houfnice, from houf, "crowd",

suggesting the cannon's use against massed enemies, and houf is in turn a borrowing from the

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Middle High German word Hūfe or Houfe (modern German Haufen), meaning "heap". Haufen,

sometimes in the compound Gewalthaufen, also designated a pike square formation in German.

Since the First World War, the word howitzer has been increasingly used to describe artillery

pieces that, strictly speaking, belong to the category of gun-howitzer - relatively long barrels

and high muzzle velocity combined with multiple propelling charges and high maximum

elevation. This is particularly true in the armed forces of the United States, where gun-

howitzers have been officially described as "howitzers" for more than sixty years. Because of

this practice, the word "howitzer" is used in some armies as a generic term for any kind of

artillery piece that is designed to attack targets using indirect fire. Thus, artillery pieces that

bear little resemblance to howitzers of earlier eras are now described as howitzers, although the

British call them guns. Most other armies in the world still reserve the word howitzer for guns

with barrel length 15 to 25 times its caliber, longer-barreled guns being cannons.

The British had a further method of nomenclature that they adopted in the 19th century. Guns

were categorized by projectile weight in pounds while howitzers were categorized by caliber in

inches. This system broke down in the 1930s with the introduction of gun-howitzers.

Current U.S. Military doctrine defines howitzers as any cannon artillery capable of high-angle

(45' to 90' elevation) and low angle (45' to 0' elevation); guns are defined as only capable of

low-angle fire and mortars are only capable of high-angle fire.

In the early 20th century the introduction of howitzers that were significantly larger than the

heavy siege howitzers of the day made necessary the creation of a fourth category, that of

"super-heavy siege howitzers". Weapons of this category include the famous Big Bertha of the

German Army and the 15-inch (381 mm) howitzer of the Royal Marine Artillery. These large

howitzers were transported mechanically rather than by teams of horses. They were transported

as several loads and had to be assembled on their firing position.

Types of Howitzer:

1. A self-propelled howitzer is mounted on a tracked or wheeled motor vehicle. In many cases,

it is protected by some sort of armor so that it superficially resembles a tank, but mostly it's not

designed for front line and cannot withstand direct anti-armor fire, instead protecting crew from

shrapnel and small arms.

2. A pack howitzer is a relatively light howitzer that is designed to be easily broken down into

several pieces, each of which is small enough to be carried by a mule or a packhorse.

5

3. A mountain howitzer is a relatively light howitzer designed for use in mountainous terrain.

Most, but not all, mountain howitzers are also pack howitzers.

4. A siege howitzer is a howitzer that is designed to be fired from a mounting on a fixed

platform of some sort.

5. A field howitzer is a howitzer that is mobile enough to accompany a field army on campaign.

It is invariably provided with a wheeled carriage of some sort.

Figure 2: HOWITZER image

6

2.2 Earlier firing methods:

Forward Observer (FO) is a vital component in the whole firing process. This observer was in

the earlier periods essentially a person which gives distance of the target for firing. Forward

observer is equipped with binoculars for viewing and compass for determining surrounding

wind velocity. These people are trained to carry out these operations efficiently. In modern

periods these observers are being replaced with powerful viewing instruments for increasing

accuracy. The whole unit carrying out these firing operations is known as the Field Artillery

Team.

Figure 3: Role of Forward Observer in firing process

Following article from the Wikipedia website and field army manual gives this information

regarding Forward Observer and Field Artillery Team:

In the land-based field artillery, the field artillery team is organized to direct and control

indirect artillery fire on the battlefield. Since World War I, to conduct indirect artillery fire,

three distinct components have evolved in this organization: the forward observer (or FO), the

fire direction center (FDC) and what is called the gun line (the actual guns themselves). On the

battlefield, the field artillery team consists of some combinations of all of these elements. In

other words there may be multiple FOs calling in fire on multiple targets to multiple FDCs and

7

any component may be in communication with some of the other elements depending on the

situational requirements.

2.2.1 Motivation:

To understand the modern field artillery team concept, it is necessary to understand that

modern artillery batteries shoot at targets measured in distances of kilometers and miles rather

than the old 18th Century concept of meters and yards, representing a hundredfold increase in

range. This dramatic range increase has been driven by the development of rifled cannons,

improvements in propellants, better communications and technical improvements in gunnery

computational abilities that have been ongoing since the end of the 19th Century. Since a

modern enemy is engaged at such great distances, in most cases, gunners no longer directly see

their targets and so they can not directly engage the enemy with observed direct fire, thus there

is a need for trained observers linked to the artillery units by modern communications to find

and adjust fire on targets at great distances. In most field artillery situations, because of

weather, terrain, night-time conditions, distance or other obstacles, the soldiers manning the

guns cannot see the target that they are firing upon. The term indirect fire is therefore used to

describe firing at targets that gunners cannot see. In most cases, the target is either over the

horizon or on the other side of some physical obstruction, such as a hill, mountain or valley.

Since the target is not visible these gunners have to rely on a trained artillery observer, also

called a forward observer, who sees the target and relays the coordinates of the target to their

fire direction center. The fire direction center, in turn, translates those coordinates into first, a

left-right aiming direction, second, an elevation angle, third, a calculated number of bags of

propellant and finally, a fuse with a determined waiting time before exploding to be set (if

necessary). The fuse is then mated to the artillery projectile.

2.2.2 Organization:

2.2.2.1 Forward Observer (FO):

Because artillery is an indirect fire weapon, the forward observer must take up a position where

he can observe the target using tools such as maps, compass, binoculars and laser

rangefinder/designators; then call back fire missions on his radio or other communication

8

device. This position can be anywhere from a few hundred meters to 20–30 km distant from the

guns. Modern day FOs are also trained in the rudiments of calling Close Air Support, Sea-

borne Weapons and other weapons systems.

Using a standardized format, the FO sends either an exact target location or the position relative

to his own location or a registered map point, a brief target description, a recommended

ammunition to use, and any special instructions such as "danger close" (The warning that

friendly troops are within a certain distance from the target, which varies based upon the

weapon system being used and which requires extra precision from the guns). Once firing

begins, if the rounds are not accurate the FO will issue instructions to adjust fire in four

dimensions (Three physical; left/right, forward/back, up/down and one for time, when using

timed fuses) and then usually call "fire for effect", unless his purpose in that fire mission has an

objective other than suppression or destruction of the target. A "Fire For Effect" or "FFE" calls

for all of the guns or tubes to fire a round; as opposed to the adjustment phase wherein only a

single gun is firing.

The FO does not talk to the guns directly - he deals solely with the FDC. The forward observer

can also be airborne and in fact one of the original roles of aircraft in the military was airborne

artillery spotting.

2.2.2.2 FDC (Fire Direction Center):

Typically, there is one FDC for a battery of six guns, in a light division. In a typical heavy

division configuration, there exist two FDC elements capable of operating two four gun

sections, also known as a split battery. The FDC computes firing data, fire direction, for the

guns. The process consists of determining the precise target location based on the observer's

location if needed, then computing range and direction to the target from the guns' location.

This data can be computed manually, using special protractors and slide rules with

precomputed firing data. Corrections can be added for conditions such as a difference between

target and howitzer altitudes, propellant temperature, atmospheric conditions, and even the

curvature and rotation of the Earth. In most cases, some corrections are omitted, sacrificing

accuracy for speed. In recent decades, FDCs have become computerized, allowing for much

faster and more accurate computation of firing data.

9

2.2.2.3 Guns:

The final piece of the puzzle is the "gun line" itself. The FDC will transmit a warning order to

the guns, followed by orders specifying the type of ammunition, fuze setting and propelling

charge, bearing, elevation, and the method of adjustment or orders for fire for effect (FFE).

Elevation (vertical direction) and bearing orders are specified in mils, and any special

instructions, such as to wait for the observer's command to fire relayed through the FDC. The

crews load the howitzers and traverse and elevate the tube to the required point, using either

hand cranks (usually on towed guns) or hydraulics (on self-propelled models).

2.2.2.4 Parent battalion and US Army brigade/USMC regimental FDCs:

FDCs also exist in the next higher parent battalion that "owns" 2-4 artillery batteries. Once

again, an FDC exists at the US Army brigade or USMC regimental level that "owns" the

battalions. These higher level FDCs monitor the fire missions of their subordinate units and

will coordinate the use of multiple batteries or even multiple battalions in what is called a

battalion or brigade/regimental mission. In training and wartime exercises, as many as 72 guns

from 3 battalions may all be coordinated to put "steel on the target" in what is called a

"brigade/regimental time on target" or brigade/regimental TOT for short. The rule is "silence is

consent," meaning that if the lower unit does not hear a "cancel the mission" (don't shoot) or

even a "check firing" (cease firing) order from the higher monitoring unit, then the mission

goes on. Higher level units monitor their subordinate unit's missions both for both active as

well as passive purposes. Higher level units also may get involved to coordinate artillery fire

across fire support coordination boundaries (often parallel lines on maps) where one unit

cannot fire into without permission from higher and/or adjacent units that "own" the territory.

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2.2.2.5 Direct fire exceptions to usual mission of artillery indirect fire:

Artillery gunners are taught how to use direct fire to engage a target such as mounted or

dismounted troops attacking them. In such a case, however, the artillery crews are able to see

what they are shooting at. With indirect fire, in normal artillery missions, the crews manning

the guns cannot see their target directly, or observers are doing that work for them. There have

been exceptions to this situation, but even when US Marines assaulted Iwo Jima during World

War Two, and gunners could see the impact of their rounds on Mt. Suribachi, the actual

adjustment of their fires was accomplished by forward observers directly supporting and

attached to infantry units, because they were in the position to see not only the enemy but to

prevent friendly fire incidents and to coordinate shelling the Japanese with their infantry unit's

movements.

2.2.3 CAPABILITIES AND LIMITATIONS:

A. The accuracy of calls for fire depends on the actions and capabilities of forward observers

(FOs) and company fire support officers (FSOs) and the accuracy of fire support plans.

B. Error-free self-location and precise target location are ideals for which the forward observer

must strive. First-round FFE on a target of opportunity and immediate and effective

suppression of enemy direct fire systems are musts if the supported maneuver unit is to

accomplish its mission. Moreover, accurate location of planned targets is imperative to

effective execution of a fire support plan. Accurate location of planned targets is possible only

if the enemy is under actual observation by a forward observer or other targeting asset. Fire

support may be indirect fire-but it must be directed!

C. Achievement of these goals is primarily situation-dependent. Accuracy of FA fires also

depends to a great extent on the skill and experience of the observer who calls for fire and the

equipment he uses for self-location and target location.

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D. The traditional forward observer, equipped with a map, compass, and binoculars, can expect

a mean target location error of about 500 meters. This is not enough for reliable first-round FFE

or target suppression; it is no better than it was in World War II. Lengthy adjustments of fire

are required to move the rounds onto the target. This wastes time and ammunition and gives the

enemy a chance to take cover or leave the area.

E. Attainable accuracy for modern observer teams (FISTs, COLTs, and AFSOs), equipped with

electronic and optical devices such as laser range finders and position-locating systems, is

considerably improved. When properly used by trained and qualified observers, these devices

enable the observer to attain first-round accuracy never before possible; but they have inherent

hazardous characteristics. Lasers are not eye-safe and can inflict severe eye injuries. Thus, their

use in training environments is severely restricted. Even in an actual conflict, care must be

taken to prevent injuring unprotected friendly troops. Eye-safe laser range finders for use in

training areas are currently under development and will be fielded when available.

Figure 4: Field Artillery Team

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2.3 Projectile and shell types:

Various types of projectile are used depending upon distance to be covered and velocity of the

projectile required. 155mm type of projectile is the most common used in the artillery and

readings for this project are of the same. Detailed information regarding this projectile type is

collected from internet and presented in the tabular form below.

2.3.1 155mm Projectile:

The projectiles of the 155mm Howitzer went by many names; shells, projos, rounds, and joes to

name a few. Just as there were many names, so there were several different types of projectiles

to accomplish the many missions the gun was expected to perform. Add several different fuzes,

and two different powders, and the 155 became a very versatile weapon. The first, and most

utilized projectile, was the M107 High Explosive round (HE). With the super quick, point

detonating fuze(PD), the round was used against personnel, and light armor. With the turn of a

screw, the fuze became delayed, and HE round could dig out bunkers, and other fortified

positions. With a Mechanical Time, or Variable Time fuze, the HE would airburst with

devastating effect to troops in the open or even in trenches and foxholes.

To mark where the explosive rounds would hit, or to indicate to a unit in the field where they

were, the M116(A1) Colored Smoke/High Cloud round was used. The smoke was bright red,

purple, or yellow. The high cloud round was white smoke, but could be used as a marker at

night as it left a stream of sparks going into the target. All of the smoke rounds used

mechanical time fuzes.

The White Phosphorus (WP), M110, was also used as a marker round. It could be fitted with

PD, VT, and MT fuzes. When the situation called for it, white phosphorus became a

devastating weapon against personnel. The thick white smoke could be used as a screen to

mask movement by troops in the field.

Illumination was a very important mission for artillery in Vietnam. To accomplish that, the

155mm howitzer used the M485(A1). Each round was capable of lighting up a 1000 yard

diameter area. The flare was ejected at 700 yards altitude, and would stay lit for at least 90

13

seconds. By keeping two flares in the air at a time, shadows were reduced. At a firing rate of

two rounds a minute, an area could be kept lit all night long.

The ICM (Improved Conventional Weapons), or Firecracker round (M449) was used for

antipersonnel missions. An airburst, base ejecting shell, it dropped sixty bomblets that bounced

up five feet in the air before going off. It proved to be extremely effective against enemy in the

open, or in positions with little, or no overhead protection.

We have included the gas round, M121A1 although it is not clear that the 1/92nd ever fired it in

anger. Delivered as an airburst, or as a point detonating round, it contained Ca, or Cs types of

"tear" gas.

Shell M # Color Stripe/ Color Lettering

Color Weight in

Pounds

High

Explosive M107 Olive Drab None Yellow 97

Gas M121A1 Gray Three Green, One

Yellow Green 100

WP M110 Lt. Green One Yellow Red 98

Smoke/HC M116(A1) Lt. Green None Black 86

Illumination M485(A1)* Olive Drab One White White 85

ICM M449 Olive Drab One Row Yellow

Diamonds Yellow 95

Table 1: Various types of projectiles

* The M485 was to be fired at a maximum charge 6, while the M485A1 could be fired at

Charge 7.

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Figure 5: Marking of 155mm HE projectile

The 155mm projectiles are the most widely used artillery round. They offer a wide range of

options on the battlefield. The 155mm howitzers are separate loading ammunition, which has

four separate components: primer, propellant, projectile, and fuze. The four components are

issued separately. Upon preparation for firing, the projectile and propellant are loaded into the

howitzer. Separate loading ammunition propellants are issued as a separate unit of issue in

sealed canisters to protect the propellant. The amount of propellant to be fired with artillery

ammunition is varied by the number of propellant increments. The charge selected is based on

the range to the target and the tactical situation.

Figure 6: 155mm HE projectile components

15

Figure 7: Charges used in 155mm projectile

Its accuracy is measured in centimeters, and its lethality is impressive. Copperhead is a cannon-

launched, 155mm artillery projectile which guides itself to a laser-designated target. The

ammunition is capable of defeating both armor and point targets at ranges of over six

kilometers, and provides the battlefield commander with the unparalleled capability of utilizing

artillery to the same effect as direct fire weapons and close air support. The system was

employed during Operation Desert Storm, during which it met with great success. Copperhead

projectiles were used to destroy observation and border guard posts and forward radar

installations during the first week of artillery attacks.

2.3.2 Shells:

A shell is a payload-carrying projectile which, as opposed to shot, contains an explosive or

other filling, though modern usage sometimes includes[citation needed] large solid projectiles

properly termed shot (AP, APCR, APCNR, APDS, APFSDS and proof shot). Solid shot may

contain a pyrotechnic compound if a tracer or spotting charge is used. Originally it was called a

"bombshell", but "shell" has come to be unambiguous in a military context. "Bombshell" is still

used figuratively to refer to a shockingly unexpected happening or revelation.

All explosive- and incendiary-filled projectiles, particularly for mortars, were originally called

grenades, derived from the pomegranate, whose seeds are similar to grains of powder. Words

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cognate with grenade are still used for artillery or mortar projectile in some European

languages.

Shells are usually large-calibre projectiles fired by artillery and combat vehicles (including

tanks), and warships. Shells usually have the shape of a cylinder topped by an ogive-shaped

nose for good aerodynamic performance, possibly with a tapering base; but some specialized

types are quite different.

Solid cannonballs (“shot”) did not need a fuse, but hollow munitions (“shells”) filled with

something such as gunpowder to fragment the ball, needed a fuse, either impact (percussion) or

time. Percussion fuses with a spherical projectile presented a challenge because there was no

way of ensuring that the impact mechanism hit the target. Therefore shells needed a time fuse

that was ignited before or during firing and burnt until the shell reached its target. Early reports

of shells include Venetian use at Jadra in 1376 and shells with fuses at the 1421 siege of St

Boniface in Corsica. These were two hollowed hemispheres of stone or bronze held together by

an iron hoop. Written evidence for early explosive shells in China appears in the early Ming

Dynasty (1368–1644) Chinese military manual Huolongjing, compiled by Jiao Yu (fl. 14th to

early 15th century) and Liu Ji (1311–1375) sometime before the latter's death, a preface added

by Jiao in 1412. As described in their book, these hollow, gunpowder-packed shells were made

of cast iron.

Shells have never been limited to an explosive filling. An incendiary shell was invented by

Valturio in 1460. The carcass was invented in 1672 by a gunner serving Christoph van Galen,

Prince Bishop of Munster, initially oblong in an iron frame or carcass (with poor ballistic

properties) it evolved into a spherical shell. Their use continued well into the 19th Century. In

1857 the British introduced an incendiary shell (Martin's) filled with molten iron, which

replaced red hot shot used against ships, most notably at Gibraltar in 1782. Two patterns of

incendiary shell were used by the British in World War 1, one designed for use against

Zeppelins.

The calibre of a shell is its diameter. Depending on the historical period and national

preferences, this may be specified in millimetres, centimetres, or inches. The length of gun

barrels for large cartridges and shells (naval) is frequently quoted in terms of the ratio of the

barrel length to the bore size, also called calibre. For example, the 16"/50 caliber Mark 7 gun is

50 calibers long, that is, 16"×50=800"=66.7 feet long. Some guns, mainly British, were

specified by the weight of their shells. Due to manufacturing difficulties the smallest shells

17

commonly used are around 20 mm calibre, used in aircraft cannon and on armored vehicles.

Smaller shells are only rarely used as they are difficult to manufacture and can only have a

small explosive charge. The largest shells ever fired were those from the German super-railway

guns, Gustav and Dora, which were 800 mm (31.5") in calibre. Very large shells have been

replaced by rockets, guided missile, and bombs and today the largest shells in common use are

155 mm (6.1"). Gun calibres have standardized around a few common sizes, especially in the

larger range, mainly due to the uniformity required for efficient military logistics. Shells of

105, 120, and 155 mm diameter are common for NATO forces' artillery and tank guns.

Artillery shells of 122, 130 and 152 mm, and tank gun ammunition of 100, 115, or 125 mm

calibre remain in use in Eastern Europe and China. Most common calibres have been in use for

many years, since it is logistically complex to change the calibre of all guns and ammunition

stores. The weight of shells increases by and large with calibre. A typical 150 mm (5.9") shell

weighs about 50 kg, a common 203 mm (8") shell about 100 kg, a concrete demolition 203 mm

(8") shell 146 kg, a 280 mm (11") battleship shell about 300 kg, and a 460 mm (18") battleship

shell over 1500 kg. The Schwerer Gustav supergun fired 4.8 and 7.1 tonne shells.

Figure 8: Some sectioned shells from World War 1

2.3.3 Fuzes:

Fuzes form an important component in the projectile. It is used in the explosion of the

projectile. The fuzes were as varied as the projectiles. Depending on which round was used,

and the mission it was to perform, several of the rounds took two or more fuzes. Here are the

fuzes, and what they were used for.

Fuze Type Uses Projectiles

M557/M739 SQPD/Delay Impact/Subsurface HE,WP,Gas

18

M564 MT/SQ Airburst/Impact HE,WP,Gas

M565 MT Airburst Illumination, M449

M501(A1) MT/SQ Airburst/Impact High Cloud/Smoke

M729 Proximity/SQ Airburst/Impact HE, WP, Gas

M78(A1) CP/Delay Concrete HE, WP, Gas

Table 2: Various types of shells and its uses

2.4 Point mass model:

Different types of model are available for computation of projectile motion with the simplest

one being vacuum model and the most complicated one being 6DOF (degree of freedoms)

model. For the sake of simplicity and required accuracy, a Point Mass Model is implemented in

this project. The differential equations regarding this model are presented with suitable

assumptions.

This model assumes that all the projectile mass is located at a single point. This model also

accounts for the drag caused by the air, but like the vacuum model, it neglects the aerodynamic

forces and moments that act on projectile. The trajectory is given by the following set of

equations:

19

Where is the „p‟ air density, „d‟ is the projectile diameter, „Cd‟ is the drag coefficient, „m‟ is the

projectile mass, is the range, „y‟ is the height, „z‟ is the drift, „Wx‟, „Wy‟ and „Wz‟ are wind

velocity for the three axes . This model is fairly accurate and has been used by manufacturers to

generate firing tables for some period of time.

Following is the article extracted from the manual of the BRL. It explains the use of differential

equations assuming point mass model for computing the trajectories of the projectile.

The equations of motion, incorporating all six degrees of freedom of a body in free flight, have

been programmed for the BRLESC and are used for the burning phase of rocket trajectories.

The procedure is a very lengthy one, however; even on the very high speed BRLESC, average

computing time is approximately 4 seconds per second of time of flight. For cannon artillery

tables, this computing time would be prohibitive. In preparing a firing table for a howitzer, we

compute about 200,000 trajectories having an average time of flight of about 50 seconds. This

would mean approximately 10,000 hours of computer time. In contrast, the equations of motion

for the particle theory, which are currently used for computing firing tables for cannon artillery,

use far less computer time: approximately 1 second per 160 seconds of time of flight. For the

same howitzer table used as an example above, approximately 20 hours of computer time are

required.

Although the trajectory computed by the particle theory does not yield an exact match along an

actual trajectory, it does match the end points. For present purposes, this theory provides the

requisite degree of accuracy for artillery firing tables.

The accelerations, velocities and positions necessary to describe the particle theory are

referenced to a ground-fixed, right hand, coordinate system. The equations of motion which are

used in the machine reduction of the fitting data are:

20

Where the dots indicate differentiation with respect to time,

1. x, y and z - distances along the x, y and z axes,

2. p - air density as a function of height,

3. V - velocity,

4. KD - drag coefficient,

5. C - ballistic coefficient,

6. Wx - range wind,

7. Wz - cross wind,

8. g - acceleration due to gravity,

9. ax, ay and az are accelerations due to the rotation of the earth.

For a given projectile, KD varies with Mach number and with angle of attack. The ballistic

coefficient, C, defined as weight over diameter squared (W/d2) is a constant. However, for

convenience in handling data along any given trajectory, KD is allowed to vary only with Mach

number, and C becomes a variable. In other words, the KD used is that for zero angle of attack.

In actual flight, drag increases with an increase in angle of departure, due to large summital

yaws at high angles. Thus, if KD is not allowed to increase with increasing angle, C will

decrease in order to maintain the correct KD /C ratio. Up to an angle of departure of 45 degree,

however, sumnital yaws are so small that C is usually a constant for any given muzzle velocity.

21

2.5 Ballistic coefficient, Drag coefficient and Mach number:

It started back around 1850 when ballisticians of many countries began experiments in an effort

to improve the accuracy of artillery shells and the measurements of the drag or air resistance

they encountered during flight. Basically, the necessities of war meant everyone wanted more

accuracy. There were no computers back in those days so all the mathematical solutions to

these very complex equations were hand written. These took months and even years to

complete. Between 1875 and 1898, German, French, Russian and English ballisticians worked

feverishly to quantify air drag resistance of artillery shells and finally came up with a standard

model of projectile on which further calculations could be based. This made it a little easier to

calculate the trajectories of new shaped projectiles by reducing the time required for new

calculations. This standard reference projectile shape is known as the G1 Standard bullet. "G"

stands for the Gâvre Commission of the French Naval Artillery. This commission conducted

many air resistance firings at the Gâvre Proving Ground utilizing a Belgian chronograph

manufactured in 1864. Figure illustrates the shape and measurement of this projectile in

"calibres". One calibre is the width of the projectile.

Figure 9: Measurement of projectile in calibres

The G1 projectile was one pound in weight (lead), and was one inch in diameter. This hefty

7000 grain (1lb) projectile is basically what most small-arms projectiles today are measured

against for their ballistic coefficient reference numbers.

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2.5.1 Drag and form factors:

The amount of drag that a projectile experiences in supersonic flight depends heavily on its

shape and velocity. The speed of sound at sea level at a 15°C and 78% relative humidity (RH)

may be around 1116fps (340m/s). This can be referred to as Mach 1. A projectile travelling at

this speed is travelling at the same speed that sound travels in the same atmosphere. A

projectile travelling in this same atmosphere at 2232fps will be doing Mach 2. Mach 2.5 would

be approximately 2790fps and so on.

A projectile travelling at these speeds has shock waves of compressed air attached to the front

and rear, which tend to draw a large amount of energy from it, thus slowing it down

aggressively. These shock waves are attached to the projectile at certain angles that change at

different speeds and, as a result, draw different amounts of energy from the projectile. What

this means is that the amount of drag or resistance on the projectile varies at different speeds.

The two main factors that affect drag (air resistance) on a spin-stabilised free-flight projectile

are shape and velocity. The blunt-nosed G1 projectile will be less efficient through the air than

the G7 as it is simply not as streamlined. The relationship of a projectile's weight and its cross-

sectional area is called the "sectional density".

The "form factor" of a projectile is a numerical figure that compares a projectile's unique drag

to that of a standard or reference bullet such as the G1 or G7 projectile. The lower the form

factor (FF) of the projectile, the more efficient it is. Comparing the G1 to the G7:

At a velocity of 2792fps the amount of drag on the G1 projectile can be quantified into a

numerical figure of, say, 0.540. At the same velocity, the G7 projectile may have a drag

coefficient of 0.270. The lower the drag coefficient (CD), the more efficient it is through the

air. Compare these two by dividing the G7 by the G1 figures and you have the G1 form factor

of 0.5. If the figure is below 1.0 the projectile is more efficient than the reference projectile.

Figure 10: G7 is twice efficient in air compared to G1

23

2.5.2 Ballistic coefficient:

A ballistic coefficient is a numerical figure usually between 0 and 1 that allows you to see

basically how well it penetrates through the air. The closer, more accurate description would be

"a numerical factor that describes the rate of velocity degradation of a particular projectile

when compared with the rate of velocity degradation of a standard projectile". This figure is

determined by two attributes of the projectile: sectional density and form factor.

The sectional density of a 210gn Berger VLD would be as follows:

Divide this number by the G1 form factor (i1), and you have the G1 ballistic coefficient. This

would read as:

If this projectile was travelling at a lower velocity, the G1 ballistic coefficient would change.

At 2000fps the FF may be around .496. This would mean the G1 BC would be:

24

You can see what is happening here. The i1 FF is changing at different speeds because it is

made from the drag coefficient. The drag coefficient is changing as the velocity is changing

(slowing down). The G1 BC given to us by Berger is the average coefficient experienced

throughout the entire supersonic flight of the projectile. In this instance, the average G1 BC of

this Berger 210gn projectile is 0.631. BCs supplied by other manufacturers may not be the

average, but ones tested at short range at one or more velocities.

Finally, following are 3 simple rules to remember:

1. The higher the ballistic coefficient, the better it slices through the air.

2. The higher the drag coefficient, the worse it slices through the air.

3. The higher the sectional density, the deeper the penetration.

25

2.5.3 Mach number:

Mach number is a dimensionless quantity and it is defined as the ratio of velocity of projectile

to the velocity of sound at standard conditions. Mathematically this quantity is represented as:

Where,

1. M is the Mach number,

2. v is the velocity of the projectile in the medium, and

3. vsound is the speed of sound in the medium.

In projectile motion this quantity changes with respect to the drag coefficient of the projectile.

Following table is taken from BRL manual which shows relationship between Mach number

and drag coefficient (KD) for the 155mm projectile „HE M101‟.

Figure 11: Mach number variation with drag coeffient for HE M101

26

2.6 Fuzzy logic basics & fuzzy systems:

Fuzzy logic can be conceptualized as a generalization of classical logic. Modern fuzzy logic

was developed by Lotfi Zadeh in the mid-1960s to model those problems in which imprecise

data must be used or in which the rules of inference are formulated in a very general way

making use of diffuse categories. In fuzzy logic, which is also sometimes called diffuse logic,

there are not just two alternatives but a whole continuum of truth values for logical

propositions. A proposition A can have the truth value 0.4 and its complement Ac the truth

value 0.5. According to the type of negation operator that is used, the two truth values must not

be necessarily add up to 1.

Fuzzy logic has a weak connection to probability theory. Probabilistic methods that deal with

imprecise knowledge are formulated in the Bayesian framework, but fuzzy logic does not need

to be justified using a probabilistic approach. The common route is to generalize the findings of

multivalued logic in such a way as to preserve part of the algebraic structure. There is a strong

link between set theory, logic, and geometry. A fuzzy set theory corresponds to fuzzy logic and

the semantic of fuzzy operators can be understood using a geometric model. The geometric

visualization of fuzzy logic will give us a hint as to the possible connection with neural

networks.

Fuzzy logic can be used as an interpretation model for the properties of neural networks, as

well as for giving a more precise description of their performance. It can be shown that fuzzy

operators can be conceived as generalized output functions of computing units. Fuzzy logic can

also be used to specify networks directly without having to apply a learning algorithm. An

expert in a certain field can sometimes produce a simple set of control rules for a dynamical

system with less effort than the work involved in training a neural network. A classical example

proposed by Zadeh to the neural network community is developing a system to park a car. It is

straightforward to formulate a set of fuzzy rules for this task, but it is not immediately obvious

how to build a network to do the same nor how to train it. Fuzzy logic is now being used in

many products of industrial and consumer electronics for which a good control system is

sufficient and where the question of optimal control does not necessarily arise.

27

2.6.1 Set concept of fuzzy logic:

The difference between crisp (i.e., classical) and fuzzy sets is established by introducing a

membership function. Consider a finite set X = {x1, x2, . . . , xn} which will be considered the

universal set in what follows. The subset A of X consisting of the single element x1 can be

described by the n-dimensional membership vector Z(A) = (1, 0, 0, . . . , 0), where the

convention has been adopted that a 1 at the i-th position indicates that xi belongs to A. The set

B composed of the elements x1 and xn is described by the vector Z(B) = (1, 0, 0, ..., 1). Any

other crisp subset of X can be represented in the same way by an n-dimensional binary vector.

But what happens if we lift the restriction to binary vectors? In that case we can define the

fuzzy set C with the following vector description:

Z(C) = (0.5, 0, 0, ..., 0)

In classical set theory such a set cannot be defined. An element belongs to a subset or it does

not. In the theory of fuzzy sets we make a generalization and allow descriptions of this type. In

our example the element x1 belongs to the set C only to some extent. The degree of

membership is expressed by a real number in the interval [0, 1], in this case 0.5. This

interpretation of the degree of membership is similar to the meaning we assign to statements

such as “person x1 is an adult”. Obviously, it is not possible to define a definite age which

represents the absolute threshold to enter into adulthood. The act of becoming mature can be

interpreted as a continuous process in which the membership of a person to the set of adults

goes slowly from 0 to 1.

There are many other examples of such diffuse statements. The concepts “old” and “young” or

the adjectives “fast” and “slow” are imprecise but easy to interpret in a given context. In some

applications, such as expert systems, for example, it is necessary to introduce formal methods

capable of dealing with such expressions so that a computer using rigid Boolean logic can still

process them. This is what the theory of fuzzy sets and fuzzy logic tries to accomplish.

28

Figure 12: Membership functions for the concepts young, mature and old

Above figure shows three examples of a membership function in the interval 0 to 70 years. The

three functions define the degree of membership of any given age in the sets of young, adult,

and old ages. If someone is 20 years old, for example, his degree of membership in the set of

young persons is 1.0, in the set of adults 0.35, and in the set of old persons 0.0. If someone is

50 years old the degrees of membership are 0.0, 1.0, 0.3 in the respective sets.

2.6.2 Fuzzy logic rules & fuzzy system:

We can introduce basic operations on fuzzy sets. Similar to the operations on crisp sets we also

want to intersect, unify and negate fuzzy sets. In his very first paper about fuzzy sets [1], L. A.

Zadeh suggested the minimum operator for the intersection and the maximum operator for the

union of two fuzzy sets. It can be shown that these operators coincide with the crisp unification,

and intersection if we only consider the membership degrees 0 and 1. For example, if A is a

fuzzy interval between 5 and 8 and B be a fuzzy number about 4 as shown in the Figure below:

Figure 13: Example fuzzy sets

29

In this case, the fuzzy set between 5 and 8 AND about 4 is:

Figure 14: Example of fuzzy operation AND (min)

Set between 5 and 8 OR about 4 is shown in the next figure:

Figure 15: Example of fuzzy operation OR (max)

The NEGATION of the fuzzy set A is shown below:

Figure 16: Fuzzy operation NEGATION

The above mentioned rules: AND (min), OR (max) and NEGATION form basics for the

IF-THEN rules which are required to construct the fuzzy system.

30

Fuzzy system consists of three components: Fuzzification, Rule evaluation & Defuzzification.

Following images shown below assists in get some idea.

Figure 17: Fuzzy system components

Figure 18: Steps involved in designing Fuzzy system

Above example shows how a fuzzy system can be designed for a simple case where tip to the

waiter of the restaurant is decided depending upon the food quality and service provided

31

Chapter 3

Present Work

The work discussed here mainly focuses on the software algorithm development for the

determination of the firing angle. Block diagram for the project is given which gives idea about

the work flow. In the next part MATLAB programs and MULTISIM circuits are discussed. The

chapter ends with the presentation of the model.

3.1 BLOCK DIAGRAM:

Figure 19: Block diagram depicting work flow

Above block shows all components of the project. Firstly MATLAB program is implemented

which ultimately gives us the angle of firing. This angle is given to angle an analog controller

which will set the howitzer to the required angle. Then it is ready to fire the projectile at desired

target. This projectile is powered by pneumatic energy which is supplied via air compressor.

32

3.2 MATLAB algorithm:

Two main algorithms are developed: Differential equations trajectory computation in

MATLAB SIMULINK modeling & fuzzy logic algorithm. Out of these two algorithms, the

signal which is forwarded to the controllers is from the fuzzy algorithm. One important part to

understand that the fuzzy logic database creation requires data from the differential equations

trajectory algorithm.

3.2.1 Differential equation trajectory algorithm:

The equations earlier discussed in the point mass model system are implemented in this

algorithm. The constant parameters taken are that of 155mm projectile which are obtained from

standard army manuals.

Here is the actual program which is implemented in the MATLAB software. Taking particular

case to demonstrate the steps involved. Following is the statement:

Fire a projectile of 155mm at 3500 meters with a wind velocity of 5 knots. Take the constants

for projectile as defined for standard 155mm one.

1. Script file to determine angle for given range in vacuum.

range=input(' enter the range : '); theta=(0.5)*asind((range)*(9.81)/(376^2)); range1=[0 range]; theta1=[0 theta];

This program gives basic angle of firing in the vacuum model which neglects air resistance and

wind drift.

33

2. For calculated angle this file computes the trajectory in wind and air drag effect.

%Constants for 155mm shell c=.152; %Drag Coefficient of a 155mm shell rho= 1.00649; %kg/m^3 (density of air) g=9.81; %m/s^2 (acceleration due to gravity) %Initial Conditions delta_t= .01; %s x(1)=0; y(1)=0; V=376; %m/s

theta2=max(theta1); u=V*cosd(theta2); v=V*sind(theta2); x=0; y=0; t(1)=0; %Start Loop i=1; while min(y)> -.001; ax=-(rho*V*c)/1362.17*(u-.5); ay=(-(rho*V*c)/1362.17)*v-g; u=u+ax*delta_t;

v=v+ay*delta_t; V=sqrt(u^2+v^2); x(i+1)=x(i)+u*delta_t+.5*ax*delta_t^2; y(i+1)=y(i)+v*delta_t+.5*ay*delta_t^2; t(i+1)=t(i)+delta_t; i=i+1; end realrange=x(1,i); realrange1=[0 realrange];

plot(x,y); xlabel('x distance (m)'); ylabel('y distance (m)'); title('Projectile Path');

This above program gives us the trajectory of the projectile in realistic conditions for the angle

calculated in vacuum model earlier.

34

Figure 20: Trajectory computation in realistic conditions

This trajectory is of angle obtained in vacuum model and its path in the air drag and wind

velocity conditions. Hence, increment of angle is required

3. Final angle of firing computation with the help of SIMULINK model.

35

Figure 21: SIMULINK model to determine angle of firing.

4. Similarly readings of vertical angle at various distances and wind velocity are taken and

compiled into excel sheet.

Figure 22: Compiled readings in excel sheet

36

3.2.2 Fuzzy logic algorithm:

1. Now using this data a fuzzy file is developed in the toolbox called fuzzy logic toolbox in the

MATLAB software. The system developed has 71 IF-THEN rules and it consists of following

2 inputs: distance of the target and wind velocity. The output of the system consists of vertical

firing angle.

Figure 23: Fuzzy toolbox in MATLAB

2. Now output angle is determined by entering input values in the program.

37

Figure 24: Fuzzy rules and output

38

Figure 25: Fuzzy rules and output

3. Next, this fuzzy system is incorporated into the SIMULINK model for convenience.

Figure 26: SIMULINK model for fuzzy system

39

4. 3-D surface displaying relation between parameters depending upon designed rules.

Figure 27: 3D surface of fuzzy system

40

3.3 Hardware implementation for the project:

The angle of firing obtained from the MATLAB algorithm is given to the analog controller

which forms set point for the barrel to position. The electrical circuitry consists of signal

conditioning circuit which performs action on the signal and amplifies suitably to drive servo

motor. This servo motor rotates the gear mechanism which rotates the barrel. Feedback is

provided for accurate positioning of the barrel.

3.3.1 Signal conditioning circuitry:

Below is the actual snapshot of the circuit as well as the simulated model in SIMULINK

software.

Figure 28: Actual signal conditioning circuitry

41

Now two snapshots are shown at two different conditions:

Figure 29: Circuit at max difference between set point & actual

Figure 30: Circuit at zero difference between set point & actual

42

3.3.2 Project model:

Figure 31: Miniature model of HOWITZER

Above image shows model in which the cannon is mounted on the wooden platform and

consists of horizontal motion. The PVC pipe barrel is capable of vertical motion with the help

of servo motor mounted on the stand constructed above plastic platform.

43

Chapter 4

Result and Discussions

Continuing with the particular example taken in the PRESENT WORK section, we compare

following thins for the taken case:

1. Differential equation trajectory algorithm output angle,

2. Fuzzy logic algorithm output angle,

3. Actual angle taken from firing table of army manual

A snapshot of the final trajectory computed with the angle of firing obtained after incrementing

the angle in the SIMULINK MODEL is given below:

Figure 32: Trajectory at required angle of projectile

We can see that the range is 3500 meters as desired. The angle of firing obtained for this

trajectory from SIMULINK model is 9.4 degree.

44

Fuzzy logic output of same case is given below:

Figure 33: Fuzzy rule viewer

We can see that the fuzzy logic algorithm gives output firing angle at 9.5 degree.

45

Finally actual reading from firing table is extracted. Below is the snapshot of firing table for

particular case taken.

Figure 34: Firing table for 155mm projectile

Angle from firing table is 9.7083degree. [For 3500m, +1 head wind factor is +3.5, hence for 5

knot wind it is 3.5*5=17.5 mils. Adding basic elevation of 155 mils, it becomes 17.5

+155=172.5 mils which is 172.5*0.05628=9.7083 degree]

46

Chapter 5

Conclusion and Future scope

Conclusion:

Following results are obtained for particular case of 155mm projectile:

1. Differential equation trajectory algorithm output angle is 9.4 degree,

2. Fuzzy logic algorithm output angle is 9.5 degree,

3. Actual angle taken from firing table of army manual is 9.7 degree.

We may conclude in the end that fuzzy logic can be successfully implemented in the firing

process of HOWITZER. It is lot easier than the formation of firing table which are not only

complicated but also time consuming. Further improvements can be made by designing more

rules in fuzzy system and implementing 6 degree of freedom model for trajectory computation.

Future scope:

1. This model can be interfaced to the MATLAB software via device like PLC making it

completely automatic in operation.

2. Many other parameters can also be included in the system like ground elevation,

velocity of the moving target.

3. Another major future scope includes inclusion of camera on the projectile. The camera

which is mounted on the projectile will capture real time image of the region once the

projectile is fired. Then the location of multiple targets can be determined and upon

giving input parameters to the algorithm designed, the HOWITZER can destroy

multiple targets simultaneously.

47

REFERENCES

1. Baranowski L., 2006, A mathematical model of flight dynamics of field artillery guided

projectiles,6th International Conference on Weaponry “Scientific Aspects of Weaponry”,

Waplewo, 44-53 [in Polish]

2. Baranowski L., 2011, Modeling, Identification and Numerical Study of the Flight

Dynamics of Ballistic Objects for the Need of Field Artillery Fire Control Systems, Military

University of Technology, Warsaw, p. 258 [in Polish]

3. Carlucci, Donald, Sidney Jacobson, (2008) Ballistics: Theory and Design of Guns and

Ammunition, CRC Press, Boca Raton, FL.

4. Lieske, R. F., and Danberg, J. E. (1992) Modified Mass Trajectory Simulation for Base-Burn

Projectiles, ADA248 292, Ballistic Research Laboratories, Aberdeen Proving Ground,

Maryland.

5. Lieske, R. F., and Reiter, M. L. (1966), Equations of Motion for a Modified Point Mass

Trajectory, Report #1314, Ballistic Research Laboratories, Aberdeen Proving Ground,

Maryland.

6. P.M. Gell, Maj. (1987), Textbook of Ballistics and Gunnery, Vol. I, Her Majesty Stationary

Office, London.

Websites:

1. en.wikipedia.org